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IOSR Journal of Electrical and Electronics Engineering (IOSR-JEEE)
e-ISSN: 2278-1676,p-ISSN: 2320-3331, Volume 10, Issue 3 Ver. III (May – Jun. 2015), PP 49-53
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Resonant Controller to Minimize THD for PWM Inverter
Atul A. Joshi 1, Ravindrakumar M. Nagarale 2
1
2
PG Student, M.B.E.Society’s College of, Engineering, Ambajogai (M.S), India
PG Department, M.B.E.Society’s College of, Engineering, Ambajogai (M.S), India
Abstract: In this paper, a method for minimization of Total Harmonic Distortion (THD) is proposed in case of
PWM inverter. Inverters are widely used in many applications such as in the UPS and ac motor drives resulting
in harmonics production. A phase compensation Resonant Controller (RSC) is proposed for 1-phase pulse width
modulation (PWM) inverter using MATLAB simulation. The simulation results are then compared with State
Feedback Controller (SFC) and State Vector Pulse Width Modulation (PWM) controller provided to approach
the system.
Keywords: Total harmonic distortion (THD), Pulse width modulation (PWM) , Resonant controller (RSC),
Feedback controller, State vector PWM controller etc.
I.
Introduction
Every wave shape has a Total Harmonic Distortion (THD) value [1]. The THD is the main criterion to
calculate the high grade performance of PWM inverters. Output voltage waveforms of inverters are distorted by
non linear loads or some uncertain conditions and hence affect the inverter output. Now days various electronic
equipments are used in various fields resulting in harmonic production & which causes low power factor,
electromagnetic interference, voltage distortion etc. Harmonic current occur in a facility‟s electrical system can
cause equipment malfunction, data distortion, transformer and motor insulation failure, overheating of 11
neutral buses, tripping of circuit breakers, and solid-state component breakdown. The reduction of harmonic
distortion and control of output waveforms of PWM converters have attracted many researchers‟ interests.
The State vector PWM technique [2-3] algorithm offers a novel method for minimizing the total
harmonic distortion (THD) of the output voltage of the inverter and uses simple arithmetic for determining the
sector and does not require lookup tables. However, due to the high order structure of its internal model, State
space vector PWM technique is found to be very slow in removing high order harmonics in case of PWM
inverter. However, State Feedback Controller (SFC) [3] is based on an accurate nominal model of the PWM
inverter. In the presence of parameter uncertainties and nonlinear loads, phase compensation MRSC 𝑅 𝑧 are
plugged into SFC controlled inverter to improve the tracking accuracy and eliminate the dominant harmonics.
As all we know, odd harmonics in the low frequency band dominate the total harmonic distortion of the output
voltage of the single phase PWM inverter with the rectifier load, especially 3rd, 5th, 7th, 9th, 11th,13th and 15th
order harmonics. However, due to the parameter uncertainties, unknown delays and nonlinear loads, accurate
transfer function of SFC system is different from 𝐻 𝑧 .
Resonant control (RSC) [4], which is based on IMP, has infinite open-loop gain at resonant frequency,
and can achieve zero steady state error control for sinusoidal signal with the frequency at resonant frequency.
Since harmonics of the output voltage/current of power converters mainly concentrate within low frequency
band, for example, 3rd, 5th, 7th, 9th, 11th, 13th and 15th order harmonic often dominate the harmonic distortion
caused by the single-phase rectifier load. Therefore a parallel combination of multiple resonant controllers
(MRSC) at dominant frequencies can be used to eliminate major harmonic distortion, and can obtain high
control accuracy. Phase compensation method is proposed for RSC to achieve wider stability range. Finally,
phase compensation resonant control scheme is applied to a single-phase PWM inverter to reduce its THD
value. There are many techniques for harmonic reduction, such as, Using Walsh function technique By using 12
pulse transformer design, Using Input Trap filter, Using PSO - Particle swarm optimization, Using Genetic
algorithms, Using Repetitive controller, By using Space vector modulation, State feedback controller, Resonant
controller. Above all techniques are used for harmonic reduction in case of inverters. But the main and
advantageous one is the Resonant controller. We can reduce THD through it.
II.
PWM Inverter
The power circuit of a single-phase PWM inverter [5-8] is shown in Fig (1). The output signal
generated by the PWM inverter is the pulse width modulated signal, having constant frequeney. It includes an
insulated gate bipolar transistor (IGBT) half bridge configuration and an filter. The equivalent series resistance
(ESR) of the filter capacitor is not considered because of its negligible effect on the system performance in the
concerned frequency region.
DOI: 10.9790/1676-10334953
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Resonant Controller To Minimize THD for PWM Inverter
Fig. (1). Power circuit of single phase inverter
The following differential equations can be written from the power circuit shown in Fig.:
𝐿
𝑑𝑖𝐿 𝑉𝑖𝑛
=
– 𝑉𝑜 − 𝑟𝐿 𝑖𝐿
𝑑𝑡
2
(1)
𝑑𝑣𝑜
= 𝑖𝐿 𝑖𝑜
𝑑𝑡
(2)
𝑖𝐶 = 𝑐
In the most straightforward implementation, generation of the desired output voltage is achieved by
comparing the desired reference waveform (modulating signal) with a high frequency triangular „carrier‟ wave.
Depending on whether the signal voltage is larger or smaller than the carrier waveform, either the positive or
negative dc bus voltage is applied at the output. Note that over the period of one triangle wave, the average
voltage applied to the load is proportional to the amplitude of the signal (assumed constant) during this period.
The resulting chopped square waveform contains a replica of the desired waveform in its low frequency
components, with the higher frequency components being at frequencies of close to the carrier frequency.
Notice that the root mean square value of the ac voltage waveform is still equal to the dc bus voltage, and hence
the total harmonic distortion is not affected by the PWM process. The harmonic components are merely shifted
into the higher frequency range and are automatically filtered due to inductances in the ac system.
Although the SPWM [3] waveform has harmonics of several orders in the phase voltage waveform, the
dominant ones other than the fundamental are of order n and n±2 where n = fc/fm. This is evident for the
spectrum for n=15 and m = 0.8 shown in Fig.5. Note that if the other two phases are identically generated but
120o apart in phase, the line-line voltage will not have any triple n harmonics. Hence it is advisable to choose
𝑓𝑐
= 3𝑘, (𝑘 ∈ 𝑛) , as then the dominant harmonic will be eliminated. The dominant 15th harmonic is effectively
𝑓𝑚
eliminated in the line voltage. Choosing a multiple of 3 is also convenient as then the same triangular waveform
can be used as the carrier in all three phases, leading to some simplification in hardware.
But high performance pulse-width modulated (PWM) inverters should accurately regulate the output ac
voltage/current to the reference sinusoidal input with low total harmonics distortion (THD) and fast dynamic
response. Nonlinear loads such as the rectifier loads that cause periodic distortion are major sources of THD.
However, ac references and disturbances contain dominant odd-harmonic frequencies and minor even-harmonic
frequencies in the PWM inverters.
III.
Resonant Controller
Resonant control (RSC), which is based on IMP, has infinite open-loop gain at resonant frequency, and
can achieve zero steady state error control for sinusoidal signal with the frequency at resonant frequency [4].
Since harmonics of the output voltage/current of power converters mainly concentrate within low frequency
band, for example, 3rd, 5th, 7th, 9th, 11th, 13th and 15th order harmonic often dominate the harmonic distortion
caused by the single-phase rectifier load. Therefore a parallel combination of multiple resonant controllers
(MRSC) at dominant frequencies can be used to eliminate major harmonic distortion, and can obtain high
control accuracy. However, in practical applications, when a RSC with high-order resonant frequency (e.g.,
seventh, ninth, eleventh order harmonics) is plugged into MRSC, the control system is apt to become unstable.
With the help of phase compensation principle for repetitive control [4], phase compensation method is
proposed for RSC to achieve wider stability range. Phase compensation at each resonant frequency enables
more RSCs to be added into MRSC. Therefore both stability and control accuracy of multiresonant control
DOI: 10.9790/1676-10334953
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Resonant Controller To Minimize THD for PWM Inverter
system can be improved. Finally, phase compensation resonant control scheme is applied to a single-phase
PWM inverter to evaluate its validity.
The phase compensation principle for Resonant controller is used for PWM inverters. In which parallel
combination of multiple resonant controllers at dominant harmonic frequencies can be used to eliminate
dominant harmonic distortion as follows:
𝑖
𝑅 𝑠 =
𝑛=1
𝑠 cos 𝜔𝑛 Ƭ𝑛 − 𝜔𝑛 sin 𝜔𝑛 Ƭ𝑛
𝑔𝑛
=
𝑠 2 + 𝜔𝑛 2
𝑖
𝑅𝑛 𝑠
(3)
𝑛=1
This principle is used to simulate the controller. Where R n (s) is a RSC with phase compensation time
τn and gain g n [5] . And, average phase of RSC at its resonant frequency is θn . Obviously, each RCS in MRSC
of (3) can choose its gain and compensation phase independently. Independent phase compensation and gain for
each RSC enable MRSC to achieve zero phase compensation at each dominant harmonic frequency (i.e.,
resonant frequency) with an optimal error convergence rate. Therefore, more RSCs can be added into MRSC to
enhance its tracking accuracy. Compared with phase compensation RC, phase compensation MRSC can achieve
almost the same tracking accuracy at much higher error convergence rate. Since MRSC is only comprised of
RSCs at the dominant harmonic frequencies in the low frequency band, it doesn‟t need a low pass filter.
IV.
Resonant Controller For PWM Inverter
Fig. (2) presents a single phase PWM inverter with its resonant control scheme [4]. The control
objective of the PWM inverter is to force output voltage Vc to track the reference input ref accurately. The
discrete mathematical model of such a single phase PWM inverter can be expressed as:
𝑓
𝑓
𝑣𝑐 𝑘 + 1
𝑣𝑐 𝑘
𝑕
= 11 12
+ 1 𝑉𝑖𝑛 𝑘
4
𝑓
𝑓
𝑕
𝑣𝑐 𝑘 + 2
21
22 𝑣𝑐 𝑘
2
𝑉𝑐 (𝑘)
𝑉𝑐 (𝑘)
𝑦 𝑘 = 1 0
Where,
𝑇2
𝑓11 = 1 − 2𝐿
𝑛 𝐶𝑛
𝑇
, 𝑓21 = −
𝑓22 = 1 −
𝑕1 =
(5)
𝑇2
2𝐿𝑛 𝐶𝑛
𝑇
𝑅𝑛 𝐶𝑛
𝐿𝑛 𝐶𝑛
−
, 𝑕2 =
+
𝑇2
2𝐿𝑛 𝐶𝑛
𝑇2
2𝐿𝑛 𝐶𝑛 𝑅𝑛 𝐶𝑛
𝑇2
+
𝑇
𝑇 1−
2𝑅 𝑛 𝐶 𝑛
,
2𝑅𝑛 𝐶𝑛 𝑅𝑛 𝐶𝑛
,
𝐿𝑛 𝐶𝑛
Fig. (2) - Single phase PWM inverter with Resonant control scheme.
DOI: 10.9790/1676-10334953
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Resonant Controller To Minimize THD for PWM Inverter
Where En , Ln , Cn and R n are the nominal value of the dc voltage, the inductance, the capacitance and
the load resistance respectively, is the output voltage Vc of the inverter, T is the sampling period.
Fig. (3) presents the resonant control system [6]-[7] of the single phase PWM inverter, where 𝑟𝑒𝑓 is the
reference input,𝑒 is the control error, 𝑢 is the control input, 𝑑 is the disturbance,𝐾(𝑧) is the SFC of (4), 𝑃(𝑧) is
the plant of (3), 𝑅(𝑧) denotes digital multiple RSCs.
Fig. (3)- Block Diagram of Resonant Control inverter system
A state feedback controller (SFC)[4] can be employed as follows:
µ 𝑘 = −𝐾𝑓 𝑥 𝑘 + 𝑕 𝑟𝑒𝑓 𝑘
(6)
To control the PWM inverter, where𝐾𝑓 = 𝐾1 𝐾2 , 𝑕 is the gain of the reference input . The poles of the
feedback control system can be assigned to achieve good robustness by tuning control gains𝐾1 and 𝐾2 .
V.
Simulation Results For THD
The simulation results can be seen on FFT analysis through discrete model. In case of PWM inverter,
the harmonic reductions can be shown by the main performance index THD. The harmonic reduction is done on
the selected sine wave. On that selected sine wave, to minimize the harmonics the Resonant controller is used.
The THD results are shown between the magnitude and the frequency of selected signal. For different harmonic
orders, we get different THD values. The bar representation of the harmonic orders are shown in fig (4).
Fig . (4) - THD values for Resonant controller
The table1 shown below make sure that, among all controllers or techniques for THD minimization, the
Resonant controller is the best and efficient controller.
DOI: 10.9790/1676-10334953
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Resonant Controller To Minimize THD for PWM Inverter
TABLE (1) . THD values.
SN.
1
2
3
CONTROLLERS / TECHNIQUES
Resonant Controller
Space vector PWM technique
State feedback Controller
VI.
THD VALUE( % )
1.77 %
108.61 %
2.12
Conclusion
The paper proposes technique for minimization of Total Harmonic Distortion. Presently no method
gives sinusoidal output waveform with zero percentage THD. The resonant controller is efficient, compared to
space vector PWM controller and state feedback controller, since it gives 1.77% THD value. In this paper the
phase compensation principle for RSC is developed. Phase compensation multiresonant control scheme provides
a high performance control solution for PWM converter systems. A MATLAB Simulation diagram and
simulation result shows that resonant controller is very efficient to minimize the THD value.
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DOI: 10.9790/1676-10334953
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